Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening
The Maxwell-Bloch equations have been intensively studied by many authors. The main results are based on the inverse scattering transform and the Marchenko integral equations. However this method is not acceptable for mixed problems. In the paper, we develop a method allowing to linearize mixed prob...
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| Цитувати: | Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening / V.P. Kotlyarov, E.A. Moskovchenko // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 3. — С. 328-349. — Бібліогр.: 24 назв. — англ. |
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| citation_txt | Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening / V.P. Kotlyarov, E.A. Moskovchenko // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 3. — С. 328-349. — Бібліогр.: 24 назв. — англ. |
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| description | The Maxwell-Bloch equations have been intensively studied by many authors. The main results are based on the inverse scattering transform and the Marchenko integral equations. However this method is not acceptable for mixed problems. In the paper, we develop a method allowing to linearize mixed problems. It is based on simultaneous spectral analysis of both Lax equations and the matrix Riemann{Hilbert problems. We consider the case of infinitely narrow spectral line, i.e., without spectrum broadening. The proposed matrix Riemann-Hilbert problem can be used for studying temporal/spatial asymptotics of the solutions of Maxwell-Bloch equations by using a nonlinear method of steepest descent.
Уравнения Максвелла-Блоха интенсивно изучаются многими авторами. Основные результаты базируются на методе обратной задачи с использованием интегральных уравнений Марченко. Однако такой метод оказался неприемлемым для смешанных задач. В данной работе мы развиваем метод, позволяющий линеаризовать смешанные задачи. Он основан на одновременном спектральном анализе обоих уравнений Лакса и матричных задачах Римана-Гильберта. Мы рассматриваем случай бесконечно узкой спектральной линии, т.е. без уширения спектра. Предлагаемые матричные задачи Римана-Гильберта будут полезны для изучения временных/пространственных асимптотик решений уравнений Максвелла-Блоха, используя нелинейный метод наискорейшего спуска.
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Journal of Mathematical Physics, Analysis, Geometry
2014, vol. 10, No. 3, pp. 328–349
Matrix Riemann–Hilbert Problems and Maxwell–Bloch
Equations without Spectral Broadening
V.P. Kotlyarov and E.A. Moskovchenko
B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv 61103, Ukraine
E-mail: kotlyarov@ilt.kharkov.ua
Received December 7, 2012, revised March 13, 2014
The Maxwell–Bloch equations have been intensively studied by many
authors. The main results are based on the inverse scattering transform and
the Marchenko integral equations. However this method is not acceptable
for mixed problems. In the paper, we develop a method allowing to linearize
mixed problems. It is based on simultaneous spectral analysis of both Lax
equations and the matrix Riemann–Hilbert problems. We consider the case
of infinitely narrow spectral line, i.e., without spectrum broadening. The
proposed matrix Riemann–Hilbert problem can be used for studying tem-
poral/spatial asymptotics of the solutions of Maxwell–Bloch equations by
using a nonlinear method of steepest descent.
Key words: nonlinear equations, Riemann–Hilbert problem, the steepest
descent method, asymptotics.
Mathematics Subject Classification 2010: 37K15, 35Q15, 35B40.
1. Introduction
The Maxwell–Bloch (MB) equations became well known after Lamb [1–4]. In
[5], Ablowitz, Kaup and Newell proposed the inverse scattering transform (IST)
to the Maxwell–Bloch equations for studying a physical phenomenon known as
self-induced transparency. A description of general solutions to the MB equations
and their classifying was done by Gabitov, Zakharov and Mikhailov in [6]. All
the authors used the IST method based on the Marchenko integral equations. In
particular, in [6], the authors gave an approximate solution of the mixed problem
to the MB equations in the domain x, t ∈ (0, L)× (0,∞). They also emphasized
that the IST method is not adopted for mixed problems. In this paper, we develop
a method allowing to linearize mixed problems in the case of infinitely narrow
c© V.P. Kotlyarov and E.A. Moskovchenko, 2014
Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations
spectral line, i.e., without spectral broadening. It is based on the simultaneous
spectral analysis of both the Lax equation, associated with the Maxwell–Bloch
equations, and the matrix Riemann–Hilbert problems. The proposed matrix
RH problems will be useful for studying the long time/long distance (x ∈ R+)
asymptotic behavior of solutions to the MB equations by using the nonlinear
method of steepest descent as, for example, in [7] and [8].
We consider the Maxwell–Bloch equations in the form given in [6]:
∂E
∂t
+
∂E
∂x
= 〈ρ〉, (1)
∂ρ
∂t
+ 2iλρ = NE , (2)
∂N
∂t
= −1
2
(E∗ρ + Eρ∗). (3)
Here ∗ denotes the complex conjugation, E = E(t, x) is a complex valued function
of the space variable x and the time t, ρ = ρ(t, x, λ) and N (t, x, λ) are complex
valued and real functions of t, x and the spectral parameter λ. The angular
brackets 〈〉 mean averaging on λ with a given weight function n(λ) > 0,
〈ρ〉 =
∞∫
−∞
ρ(t, x, λ)n(λ)dλ,
∞∫
−∞
n(λ)dλ = 1. (4)
Equations (1)–(4) can be found in a number of physical models. One of
the most important is a model of the propagation of electromagnetic waves in
a medium with distributed two-level atoms. In particular, there are models of
self-induced transparency [5, 9], and quantum laser amplifier [10, 11]. For these
models, E(t, x) is the complex valued envelope of electromagnetic wave of a fixed
polarization, N (t, x, λ) and ρ(t, x, λ) are entries of the density matrix of the atom
subsystem
ρ̂(t, x, λ) =
(N (t, x, λ) ρ(t, x, λ)
ρ∗(t, x, λ) −N (t, x, λ)
)
. (5)
The parameter λ denotes a deviation of the passage frequency from its mean
value, and the function n(λ) characterizes the inhomogeneous broadening, i.e., it
describes the form of a spectral line. For short reviews on the MB equations see
[5, 6, 9].
We restrict ourselves to the case of the infinitely narrow spectral line where
n(λ) = δ(λ). Then 〈ρ〉 = ρ, and (1)–(3) takes the form
∂E
∂t
+
∂E
∂x
= ρ,
∂ρ
∂t
= NE ,
∂N
∂t
= −1
2
(E∗ρ + Eρ∗). (6)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 329
V.P. Kotlyarov and E.A. Moskovchenko
The mixed problem for the Maxwell–Bloch equations is defined by the following
initial and boundary conditions:
E(0, x) = E0(x), ρ(0, x) = ρ0(x), N (0, x) = N0(x), E(t, 0) = E1(t), (7)
where x ∈ (0, L) (L ≤ ∞) and t ∈ R+. The function E1(t) is a Schwartz type
function (smooth and fast decreasing at infinity). The functions E0(x), ρ0(x),
N0(x) are smooth or Schwartz type functions if x ∈ R+. If one deals with a
solution on the whole t-line, then the input pulse E1(t) should be given for t ∈ R,
and the functions ρ(t, x), N (t, x) should be given as t → −∞.
The functions ρ(t, x), N (t, x) are not independent. Indeed, Eqs. (2) and (3)
give
∂
∂t
(|ρ(t, x)|2 +N (t, x)
)
= 0,
and we put
|ρ(t, x)|2 +N (t, x) ≡ 1.
Thus we must define ρ(0, x) and
N (0, x) = ∓
√
1− |ρ(0, x)|2.
We choose the sign ”minus” to have a stable medium, the so-called attenuator ( for
example, in the model of self-induced transparency). The sign ”plus” corresponds
to an unstable medium, i.e., to a quantum laser amplifier.
Assuming that the medium is stable, we consider the functions E(t, x), ρ(t, x)
and N (t, x) which satisfy the MB Eqs. (6) in the domain x, t ∈ (0, L) × (0,∞).
We develop the IST method in the form of the matrix Riemann–Hilbert problem
in the complex z-plane and give an integral representation for E(t, x) through
the solution of a singular integral equation which is equivalent to the matrix RH
problem. This RH problem is produced by spectral functions defined via given
initial and boundary conditions for MB equations. Further, we give a formulation
of a more general matrix RH problem which has a unique solution. We prove that
the RH problem generates a compatibility system of differential equations which
is the AKNS linear Eqs. [9] for the MB equations without spectral broadening.
Thus this RH problem generates different solutions to the MB equations. Among
them there are the solutions on the whole t-axis, the solutions of the mixed
problem in the quarter tx-plane with vanishing at infinity or some periodic in t
input pulse E(t, 0).
Our approach differs from that considered in [12] for the Goursat problem
to the MB equations where a linear (but complicated) ”evolution” in x takes
place for scattering data. We develop the approach of simultaneous spectral
analysis proposed in [13–16] and in [17–20], and prove that the mixed problem is
completely linearizable by the appropriate matrix RH problem.
330 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations
2. Basic Solutions of the Ablowitz–Kaup–Newel–Segur Linear
Equations
We use the simultaneous spectral analysis of the linear t-equation
Φt + iλσ3Φ = −H(t, x)Φ, (8)
σ3 =
(
1 0
0 −1
)
, H(t, x) =
1
2
(
0 E(t, x)
−E∗(t, x) 0
)
and the linear x-equation
Φx − i
(
λσ3 +
F (t, x))
4λ
)
Φ = H(t, x)Φ, (9)
F (t, x) =
(N (t, x) ρ(t, x)
ρ∗(t, x) −N (t, x)
)
.
Here Φ(t, x, λ) is a 2 × 2 matrix-valued function and λ ∈ R is a parameter. It
is easy to verify that the over-determined system of differential Eqs. (8), (9)
(AKNS system of equations [9]) is compatible if and only if the functions E(t, x),
ρ(t, x, ) and N (t, x) satisfy the MB Eqs. (6).
Let us rewrite equations (8) and (9) in the form:
Wt = U(t, x, λ)W, (10)
Wx = V (t, x, λ)W, (11)
where U and V are the matrices:
U(t, x, λ) =− (iλσ3 + H(t, x)),
V (t, x, λ) =iλσ3 + H(t, x) +
iF (t, x)
4λ
.
Lemma 2.1. Let Eqs. (10) and (11) be compatible for all t, x, λ ∈ R. Let
W (t, x, λ) be a matrix satisfying the t-equation (10) for all x (the x-equation
(11) for all t). Assume that W (t0, x, λ) satisfies the x-equation (11) for some
t = t0 ≤ ∞ (the t-equation (10) for some x = x0 ≤ ∞). Then W (t, x, λ) satisfies
the x-equation (11) for all t (satisfies the t-equation (10) for all x).
P r o o f. See, for example, in [17] (Lemma 2.1).
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 331
V.P. Kotlyarov and E.A. Moskovchenko
2.2. The problem on whole t-line
Let E(t, x), ρ(t, x) and N (t, x) be a smooth solution to the MB Eqs. (6).
We suppose additionally that E(t, x) and ρ(t, x) are fast decreasing as t → −∞
(hence N (t, x) → −1 as t → −∞). This problem was first studied in [5] by using
the inverse scattering transform and the Marchenko integral equation. Below we
give the formulation of the corresponding matrix Riemann–Hilbert problem.
Let Y (t, x, λ) be a product of the matrices
Y (t, x, λ) = W (t, x, λ)Φ(t, λ), (12)
where W (t, x, λ) satisfies the x-equation for all t and W (t, 0, λ) = I, and Φ(t, λ)
satisfies the t-equation for x = 0 under the initial condition lim
t→∞Φ(t, λ)eiλtσ3 = I.
Then, due to Lemma 2.1, the matrix Y (t, x, λ) is a compatible solution of the
Ablowitz–Kaup–Newel–Segur (AKNS) system of equations (8)–(9).
Let Z(t, x, λ) be a compatible solution of the AKNS system of equations (8)–
(9) such that
Z(t, x, λ) = Ψ(t, x, λ)W∞(x, λ), (13)
where Ψ(t, x, λ) satisfies the t-equation for all 0 ≤ x ≤ L ≤ ∞ under the initial
condition lim
t→−∞Ψ(t, x, λ)eiλtσ3 = I, and W∞(x, λ) satisfies the x-equation whose
coefficients are constant matrices as t → −∞. It is normalized by the initial
condition W∞(0, λ) ≡ I. It is easy to see that W (t, x, λ) is the fundamental
solution of the x-equation with fixed t ∈ R. The matrices Φ(t, λ) and Ψ(t, x, λ)
are the Jost solutions of the t-equation with any fixed x.
Lemma 2.2. Let E(t, x), ρ(t, x) and N (t, x) be smooth solutions to the MB
equations (6) such that E(t, x) and ρ(t, x) are fast decreasing as t → −∞. Let
E(t, 0) = E1(t) be smooth and fast decreasing as |t| → ∞. Then the Jost solutions
Φ(t, λ) and Ψ(t, x, λ) have the integral representations:
Φ(t, λ) = e−iλtσ3 +
∞∫
t
K+(t, τ, 0)e−iλτσ3dτ, Imλ = 0. (14)
Ψ(t, x, λ) = e−iλtσ3 +
t∫
−∞
K−(t, τ, x)e−iλτσ3dτ, Imλ = 0. (15)
The kernels K±(t, τ, .) satisfy the symmetry condition
K±∗(t, τ, .) = ΛK±(t, τ, .)Λ−1 with the matrix Λ =
(
0 1
−1 0
)
, and
[σ3,K
±(t, t, .)] = ±σ3H(t, .).
The kernels K±(t, τ, .) are smooth and fast decreasing as t + τ → ±∞.
332 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations
The proof of this lemma is well known (cf. [21]). It is also well known that
the vectors Φ[1](t, λ) and Φ[2](t, λ) of the matrix Φ(t, λ) = (Φ[1](t, λ), Φ[2](t, λ))
have the analytic continuations Φ[1](t, z) and Φ[2](t, z) to the lower and upper
half-planes of the complex z-plane (z = λ+iν), respectively. The vector columns
Ψ[1](t, x, λ) and Ψ[2](t, x, λ) of the matrix Ψ(t, x, λ) = (Ψ[1](t, x, λ), Ψ[2](t, x, λ))
have the analytic continuations Ψ[1](t, x, z) and Ψ[2](t, x, z) to the upper and
lower half-planes of the complex plane.
If H(t, x) ≡ 0 and F (t, x) ≡ −σ3, then the x-equation has an exact solution
eixη(λ)σ3 , where η(λ) = λ− 1
4λ . Taking this into account, we find that the function
W∞(x, λ) = eiη(λ)xσ3 , and the function Ŵ (t, x, λ) = e−iη(λ)xσ3W (t, x, λ) must
satisfy the integral equation
Ŵ (t, x, λ) = I +
x∫
0
e−iyη(λ)σ3(H(t, y) + i(F (t, y) + σ3)/4λ))eiyη(λ)σ3Ŵ (t, y, λ)dy.
(16)
The equation yields that W (t, x, λ) has an analytic continuation to the punctured
complex plane C \ {0}.
Lemma 2.3. Let E(t, x), N (t, x), ρ(t, x) be smooth functions. Then the solu-
tion W (t, x, λ) can be represented in the form:
W (t, x, λ) = eiη(λ)xσ3Ŵ (t, x, λ), (17)
where Ŵ (t, x, λ) is the unique solution of the Volterra integral equation (16).
The solution W (t, x, λ) is smooth in t and x, and it has an analytic continuation
W (t, x, z) where z = λ + iν ∈ C \ {0}. Moreover, the matrix W (t, x, z)e−iη(z)x is
continuous, bounded in C− ∪ R and has the asymptotics
W (t, x, z)e−iη(z)x =
(
1 0
0 e−2izx
)
+ O(z−1), Im z ≤ 0, z →∞,
and the matrix W (t, x, z)eiη(z)x is continuous, bounded in C+ ∪ R and has the
asymptotics
W (t, x, z)eiη(z)x =
(
e2izx 0
0 1
)
+ O(z−1), Im z ≥ 0, z →∞,
where the symbol O(.) means a matrix whose entries have the indicated order.
P r o o f. The solvability of the Volterra integral Eq. (16) and the smoothness
of the solution with respect to t and x can be easily proved by using the method
of successive approximations. Since the left-hand side
izσ3 + H(t, x) + i(F (t, x) + σ3)/4z
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 333
V.P. Kotlyarov and E.A. Moskovchenko
of the x-equation is analytic for z 6= 0, the matrix W (t, x, λ) has an analytic
continuation for z = λ + iν ∈ C \ {0}, which we denote as W (t, x, z). Thus the
matrix W (t, x, z)e−iη(z)x (W (t, x, z)eiη(z)x) is analytic, continuous for z 6= 0 in
C− ∪R (C+ ∪R) and has the above asymptotics. Moreover, they are bounded as
z → 0 and, hence, we find (determine) that they are continuous also at the point
z = 0.
Formulas (12), (14), (17) and Lemmas 2.2, 2.3 imply the following properties
of the matrix Y (t, x, λ) = (Y [1](t, x, λ) Y [2](t, x, λ)):
1) Y (t, x, λ) (λ 6= 0) satisfies the t- and x-equations (8)–(9);
2) Y (t, x, λ) = ΛY ∗(t, x, λ)Λ−1, λ ∈ R \ {0}, where Λ =
(
0 1
−1 0
)
;
3) detY (t, x, λ) ≡ 1, λ ∈ R \ {0};
4) the map (x, t) 7−→ Y (t, x, λ) (λ 6= 0) is smooth in t and x;
5) the vector column Y [1](t, x, λ) has the analytic continuation Y [1](t, x, z) for
z ∈ C−, and Y [1](t, x, z)eizt−ixη(z) is continuous in z ∈ C− ∪ R, and
Y [1](t, x, z)eizt−ixη(z) =
(
1
0
)
+ O(z−1), z →∞;
6) the vector column Y [2](t, x, λ) has the analytic continuation Y [2](t, x, z) for
z ∈ C+, and Y [2](t, x, z)e−izt+ixη(z) is continuous in z ∈ C+ ∪ R, and
Y [2](t, x, z)e−izt+ixη(z) =
(
1
0
)
+ O(z−1), z →∞.
These asymptotics follow from the formulas:
Φ[1](t, z)eizt =
(
1
0
)
+ O(z−1), Im z < 0, z →∞,
Φ[2](t, z)e−izt =
(
0
1
)
+ O(z−1), Im z > 0, z →∞,
and
W [1](t, x, z)e−iη(z)x =
(
1
0
)
+ O(z−1), Im z < 0, z →∞,
W [2](t, x, z)eiη(z)x =
(
0
1
)
+ O(z−1), Im z > 0, z →∞.
Taking into account that W∞(x, λ) = eiη(λ)xσ3 and Eqs. (13), (15), we find
that:
1) Z(t, x, λ) (λ 6= 0) satisfies the t- and x-equations (8)–(9);
2) Z(t, x, λ) = ΛZ∗(t, x, λ)Λ−1, λ ∈ R \ {0};
334 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations
3) detZ(t, x, λ) ≡ 1, λ ∈ R \ {0};
4) the map (x, t) 7−→ Z(t, x, λ) (λ 6= 0) is smooth in t and x;
5) the vector column Z[1](t, x, λ) has the analytic continuation Z[1](t, x, z) for
z ∈ C+, and Z[1](t, x, z)eizt−ixη(z) is continuous in z ∈ C+ ∪ R, and
Z[1](t, x, z)eizt−ixη(z) =
(
1
0
)
+ O(z−1), z →∞;
6) the vector column Z[2](t, x, λ) has the analytic continuation Z[2](t, x, z) for
z ∈ C−, and Z[2](t, x, z)e−izt+ixη(z) is continuous in z ∈ C− ∪ R, and
Z[2](t, x, z)e−izt+ixη(z) =
(
1
0
)
+ O(z−1), z →∞.
Since the matrices Y (t, x, λ) and Z(t, x, λ) are the solutions of the t- and
x-equations (8)–(9), they are linearly dependent. Thus, there exists a transition
matrix T (λ), independent of x and t, such that
Y (t, x, λ) = Z(t, x, λ)T (λ). (18)
The transition matrix is equal to
T (λ) = Z−1(0, 0, λ)Y (0, 0, λ) = Ψ−1(0, 0, λ)Φ(0, λ)
and, hence, T (λ) = ΛT ∗(λ)Λ−1, i.e., T (λ) has the form
T (λ) =
(
a(λ) b(λ)
−b(λ) a(λ)
)
, a(λ) = a∗(λ), b(λ) = b∗(λ).
The scattering relation (18) can be written in the following form:
Y [1](t, x, λ) = a(λ)Z[1](t, x, λ)− b(λ)Z[2](t, x, λ), λ ∈ R,
Y [2](t, x, λ) = a(λ)Z[2](t, x, λ) + b(λ)Z[1](t, x, λ), λ ∈ R.
These relations give
a(λ) = det(Z[1](t, x, λ), Y [2](t, x, λ)), b(λ) = det(Y [2](t, x, λ), Z[2](t, x, λ)).
The matrices Ψ(0, 0, λ) and Φ(0, λ) have the form
(
α(λ) −β(λ)
β(λ) α(λ)
)
and
(
A(λ) B(λ)
−B(λ) A(λ)
)
,
respectively. The functions α(λ), β(λ) and A(λ), B(λ) have analytic continua-
tions in C+ and α(z) = α∗(z∗), β(z) = β∗(z∗), and A(z) = A∗(z∗), B(z) = B∗(z∗)
are analytic in C−. Thus we have
a(λ) = α(λ)A(λ)− β(λ)B(λ), b(λ) = α∗(λ)B(λ) + β∗(λ)A(λ).
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 335
V.P. Kotlyarov and E.A. Moskovchenko
The function a(λ) has the analytic continuation a(z) for z ∈ C+, and a(z) =
a∗(z∗) is analytic in C−. The functions b(λ) and b(λ) = b∗(λ) are defined for λ ∈ R
only. The determinant of T (λ) ≡ 1 for Imλ = 0 and, hence, |a(λ)|2 + |b(λ)|2 ≡ 1.
The spectral functions have the following asymptotics: a(z) = 1 + O(z−1) as
z →∞ and Im z ≥ 0, and b(λ) = O(λ−1) as λ →∞.
If the function a(z) has zeroes zj ∈ C+, then
a(zj) = det(Z[1](t, x, zj), Y [2](t, x, zj)) = 0, j = 1, 2, ...., p.
Hence the vector columns of the determinant are linearly dependent:
Y [2](t, x, zj) = γjZ[1](t, x, zj), γj =
B(zj)
α(zj)
=
A(zj)
β(zj)
, j = 1, 2, . . . , p.
(19)
At the conjugated points z∗j ∈ C− (j = 1, 2, . . . , p), the function
a(z∗j ) = det(Y [1](t, x, z∗j ), Z[2](t, x, z∗j )) = 0.
Therefore,
Y [1](t, x, z∗j ) = γjZ[2](t, x, z∗j ), γj =
B∗(z∗j )
α∗(z∗j )
=
A∗(z∗j )
β∗(z∗j )
= γ∗j . (20)
Let us define the matrix
M(t, x, z) =
(
Z[1](t, x, z)eizt−ixη(z) Y [2](t, x, z)
a(z)
e−izt+ixη(z)
)
, z ∈ C+
(
Y [1](t, x, z)
a(z)
eizt−ixη(z) Z[2](t, x, z)e−izt+ixη(z)
)
, z ∈ C−.
(21)
The matrix is analytic for z ∈ C \ R if a(z) 6= 0, it has a jump across the real
λ-axis M(t, x, λ− i0) = M(t, x, λ + i0)J(t, x, λ), where
J(t, x, λ) =
1 + |r(λ)|2 −r(λ)e−2iθ(t,x,λ)
−r(λ)e2iθ(t,x,λ) 1
, θ(t, x, λ) = izt− iη(z)x,
(22)
and det M(t, x, λ − i0) = detM(t, x, λ + i0) = 1. The matrix M(t, x, z) has
the asymptotics M(t, x, z) = I + O(z−1) as z → ∞. If a(z) has zeroes, then
the matrix is a meromorphic function and residues relations must be added.
Namely, if the number of zeroes is finite and they are simple, i.e., a(zj) = 0 and
ȧ(zj) = da(z)/dz|z=zj 6= 0 (j = 1, 2, . . . , p), then
res
z=zj
M [2](t, x, z) = mje
−2izjt+2ixη(zj)M [1](t, x, zj), (23)
336 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations
res
z=z∗j
M [1](t, x, z) = m∗
je
2iz∗j t−2ixη(z∗j )M [2](t, x, z∗j ), (24)
where mj = γj/ȧ(zj), m∗
j = γj/ȧ
−(z∗j ), and the numbers γj , γj = γ∗j are defined
in (19) and (20).
2.2. Mixed problem
We consider here the mixed problem (6)–(7) in the domain t ∈ R+, 0 ≤
x ≤ L ≤ ∞. Let E(t, x), N (t, x), ρ(t, x) be a smooth solution of the mixed
problem. We use the compatible solution Y (t, x, λ) ( with the restriction t ∈ R+)
of the AKNS system of equations introduced in the previous subsection. Another
compatible solution Ẑ(t, x, λ) of equations (8)–(9) is defined as follows:
Ẑ(t, x, λ) = Ψ̂(t, x, λ)w(x, λ), (25)
where Ψ̂(t, x, λ) satisfies the t-equation for all x and Ψ̂(0, x, λ) = I, and w(x, λ)
satisfies the x-equation with t = 0 under the initial condition w(L, λ) = eiLη(λ)σ3
or lim
x→∞w(x, λ)e−ixη(λ)σ3 = I if L = ∞.
Lemma 2.4. Let E(t, x), N (t, x), ρ(t, x) be smooth. The function Ψ̂(t, x, λ)
has an integral representation
Ψ̂(t, x, λ) = e−iλtσ3 +
t∫
−t
L(t, τ, x)e−iλτσ3dτ. (26)
The kernel L(t, τ, x) is smooth, it satisfies the symmetry condition L∗(t, τ, x) =
ΛL(t, τ, x)Λ−1 with the matrix Λ =
(
0 1
−1 0
)
, and
[σ3, L(t, t, x)] = H(t, x)σ3.
The proof of this lemma can be found in (cf. [17]). Integral representation (26)
gives the analyticity of the Jost solution Ψ̂(t, x, z) for z ∈ C and its asymptotic
(as z →∞) behavior:
Ψ̂(t, x, z)eizt =
(
1 0
0 e2izt
)
+ O(z−1) + O(e2iztz−1),
Ψ̂(t, x, z)e−izt =
(
e−2izt 0
0 1
)
+ O(z−1) + O(e−2iztz−1),
where the symbol O(.) means a matrix whose entries have the indicated order.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 337
V.P. Kotlyarov and E.A. Moskovchenko
The function ŵ(x, λ) = w(x, λ)e−iη(λ)xσ3 must satisfy the integral equation
ŵ(x, λ) = I +
L∫
x
ei(x−y)η(λ)σ3Ĥ(y, λ)ŵ(y, λ)e−i(x−y)η(λ)σ3dy, (27)
where Ĥ(y, λ) = H(0, y)+i(F (0, y)+σ3)/4λ. By using this integral equation, we
can prove that the matrix w(x, λ) is analytic everywhere and has the asymptotic
behavior
w(x, λ) =
(
eixη(λ) 0
0 e−ixη(λ)
)
+
(
χ11(x, λ)eixη(λ) χ12(x, λ)e−ixη(λ)
χ21(x, λ)eixη(λ) χ22(x, λ)e−ixη(λ)
)
,
where χ11(x, λ), χ21(x, λ) = O(λ−1) + O(λ−1e2iLλ) and χ12(x, λ), χ22(x, λ) =
O(λ−1)+O(λ−1e−2iLλ) as λ → ±∞ (0 ≤ x ≤ L < ∞). If L = ∞, then χij(x, λ) =
O(λ−1) as λ → ±∞. Due to the analytic continuation of the first vector column
to the upper half-plane and the second vector column to the lower half-plane,
we have (χ11(x, z), χ21(x, z)) = O(z−1) for Im z ≥ 0 and (χ21(x, z), χ22(x, z)) =
O(z−1) for Im z ≤ 0 as z →∞.
Formulas (25), (26), (27) and Lemmas 2.2, 2.4 imply the following properties
of the matrices Ẑ(t, x, λ) = (Ẑ[1](t, x, λ) Ẑ[2](t, x, λ)):
1) Ẑ(t, x, λ) (λ 6= 0) satisfies the t- and x-equations (8)–(9);
2) Ẑ(t, x, λ) = ΛẐ∗(t, x, λ∗)Λ−1, λ ∈ C \ {0}, where Λ =
(
0 1
−1 0
)
;
3) det Ẑ(t, x, λ) ≡ 1, λ ∈ C \ {0};
4) the map (x, t) 7−→ Ẑ(t, x, λ) (λ 6= 0) is smooth in t and x;
5) the map z 7−→ Ẑ[1](t, x, z) is analytic in z ∈ C+;
6) the map z 7−→ Ẑ[2](t, x, z) is analytic in z ∈ C−;
7) the vector functions Ẑ[1](t, x, λ)eiλt−ixη(λ), Ẑ[2](t, x, z)e−iλt+ixη(λ) are analytic,
bounded in z ∈ C±, continuous up to the boundary (R), and
Ẑ[1](t, x, z)eizt−ixη(z) =
(
1
0
)
+ O(z−1), z ∈ C+, z →∞;
Ẑ[2](t, x, z)e−izt+ixη(z) =
(
0
1
)
+ O(z−1), z ∈ C−, z →∞.
Since the matrices Y (t, x, λ), Ẑ(t, x, λ) are the solutions of Eqs. (8)–(9), they
are linearly dependent. Thus, there exists a transition matrix T (λ), independent
of x and t, such that
Y (t, x, λ) = Ẑ(t, x, λ)T (λ). (28)
The transition matrix is equal to
T (λ) = Ẑ−1(0, 0, λ)Y (0, 0, λ) = w−1(0, λ)Φ(0, λ)
338 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations
and, hence, T (λ) = ΛT ∗(λ)Λ−1, i.e., T (λ) has the form
T (λ) =
(
a(λ) b(λ)
−b(λ) a(λ)
)
.
For the convenience of further formulation of the RH problem, we use the same
notations as in the previous subsection for the transition matrix T (λ) and its
entries. It is easy to see that the matrix w(0, λ) = ŵ(0, λ) =
(
α(λ) −β(λ)
β(λ) α(λ)
)
is
the spectral function of the x-equation for t = 0. It is uniquely defined by the
given initial functions E(0, x), ρ(0, x) and N (0, x). The function Φ(0, λ) is the
spectral function of the t-equation for x = 0, uniquely defined by the boundary
condition E(t, 0), and it has the form Φ(0, λ) =
(
A(λ) B(λ)
−B(λ) A(λ)
)
. The functions
α(λ), β(λ) and A(λ), B(λ) have analytic continuations in the upper half-plane
C+, and α(λ) = α∗(λ), β(λ) = β∗(λ) and A(λ) = A∗(λ), B(λ) = B∗(λ) have
analytic continuations in the lower half-plane C−. Thus we have
a(z) = α(z)A(z)− β(z)B(z), z ∈ C+ ∪ R;
b(λ) = α∗(λ)B(λ) + β∗(λ)A(λ), λ ∈ R.
The function a(z) is analytic in C+, and a(z) = a∗(z∗) is analytic in C−. The
functions b(λ) and b(λ) = b∗(λ) are defined for λ ∈ R only. The determinant of
T (λ) ≡ 1 for Imλ = 0 and, hence |a(λ)|2 + |b(λ)|2 ≡ 1. The spectral functions
have the following asymptotics:
a(z) = 1 + O(z−1), z →∞, b(λ) = O(λ−1), λ →∞.
If a(z) (a(z)) has zeroes, then we have the relation between the vector columns
Ẑ[1](t, x, zj) and Y [2](t, x, zj) (Y [1](t, x, z∗j ) and Ẑ[2](t, x, z∗j )) (j = 1, 2, . . . , p)
similarly to (19), (20).
Further, the matrix
M(t, x, z) =
(
Ẑ[1](t, x, z)eizt−ixη(z) Y [2](t, x, z)
a(z)
e−izt+ixη(z)
)
, z ∈ C+
(
Y [1](t, x, z)
a(z)
eizt−ixη(z) Ẑ[2](t, x, z)e−izt+ixη(z)
)
, z ∈ C−
(29)
and the scattering relation (28) generate explicitly the same matrix RH problem
as in (21), (22), (23), (24).
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 339
V.P. Kotlyarov and E.A. Moskovchenko
3. Matrix Riemann–Hilbert Problems
In this section we give a reconstruction of the solution to the MB equations in
terms of the spectral functions a(λ), b(λ), which are defined through the spectral
functions A(λ), B(λ) and α(λ), β(λ).
In the previous section we have proved that the matrices (21), (29) (due to
the scattering relations (18), (28)) are the solutions of the following matrix RH
problem RHtx:
Find the 2× 2 matrix M(t, x, z) such that
• M(t, x, z) is analytic ( if a(z) 6= 0) or meromorphic (if a(zj) = a(z∗j ) =
0, Im zj > 0, j = 1, 2, . . . , p) in z ∈ C \ R and continuous up to the real
λ-axis; RH1
• If a(zj) = a(z∗j ) = 0, j = 1, 2, . . . , p, then M(x, t, z) has poles at the points
z = zj , z = z∗j (j = 1, 2, . . . , p), and the corresponding residues satisfy the
relations:
res
z=zj
M [2](t, x, z) = mje
−2izjt+2ixη(zj)M [1](t, x, zj) RH2
res
z=z∗j
M [1](t, x, z) = m∗
je
2iz∗j t−2ixη(z∗j )M [2](t, x, z∗j ), RH3
where mj = γj/ȧ(zj), m∗
j = γj/ȧ(z∗j ), and the numbers γj , γj = γ∗j are
defined in (19) and (20);
• M−(t, x, λ) = M+(t, x, λ)J(t, x, λ), λ ∈ R, RH4
J(t, x, λ) =
1 + |r(λ)|2 −r(λ)e−2iθ(t,x,λ)
−r(λ)e2iθ(t,x,λ) 1
, λ ∈ R, (30)
where r(λ) = b(λ)/a(λ) and θ(t, x, λ) = λt− xη(λ).
• M(t, x, z) = I + O(z−1), |z| → ∞. RH5
Taking into account the well-known fact that a(z) can have multiple zeros or
infinitely many zeros with limit points on the real λ- axis or real zeroes (the so-
called spectral singularities), we propose below a more convenient formulation of
the matrix RH problem. We introduce once more the matrix solution Z0(t, x, λ)
of the AKNS system of equations normalized by the condition Z0(0, 0, λ) = I.
It is seen that Z0(t, x, λ) = Ψ̂(t, x, λ)W (0, x, λ), where Ψ̂(t, x, λ) and W (t, x, λ)
were defined in the first section. Thus, Z0(t, x, λ) is analytic in z ∈ C \ {0},
Z0(t, x, λ)eiθ(t,x,λ)σ3 is bounded in any disk |z| ≤ R. Hence the matrix can be
340 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations
extended up to the continuous function in the disk. Then we put
M̂(t, x, z) =
{
M(t, x, z), |z| > R
Z0(t, x, λ)eiθ(t,x,λ)σ3 , |z| < R,
(31)
where R is positive and sufficiently large such that a(z) 6= 0 when |z| > R. Let
the contour ΣR = (−∞,−R]∪ [R,∞)∪ΓR, where ΓR = {z : |z| = R} is the circle
of radius R oriented clockwise. Then we obtain the equivalent RH problem:
• M̂(t, x, z) is analytic in z ∈ C \ ΣR and continuous up to the contour ΣR;
RRH1
• M̂−(t, x, z) = M̂+(t, x, z)J(t, x, z), z ∈ ΣR, RRH2
J(t, x, z) =
1 + |r(λ)|2 −r(λ)e−2iθ(t,x,λ)
−r(λ)e2iθ(t,x,λ) 1
, z = λ ∈ R \ (−R, R),
α(z) B(z)e−2iθ(t,x,z)/a(z)
β(z)e2iθ(t,x,z) A(z)/a(z)
−1
, z ∈ ΓR ∩ C+,
A(z)/a(z) −β(z)e−2iθ(t,x,z)
−B(z)e2iθ(t,x,z)/a(z) α(z)
, z ∈ ΓR ∩ C−.
• M̂(t, x, z) = I + O(z−1), |z| → ∞. RRH3
We will prove now the following theorem.
Theorem 3.1. Let the functions E(t, x), N (t, x) and ρ(t, x) be the solutions
to the Maxwell–Bloch equations (1)–(3) considered in the subsection 2.1 or 2.2.
There exists the matrix M(t, x, z) which is the solution of the Riemann–Hilbert
problem (RRH1)–(RRH3), and the complex electric field envelope E(t, x) is de-
fined by the relation
E(t, x) =− lim
z→∞ 4izM12(t, x, z). (32)
The entries N (t, x) and ρ(t, x) of the matrix F (t, x) are defined as follows:
F (t, x) = −m0(t, x)σ3m
−1
0 (t, x), m0(t, x) = lim
z→0
M(t, x, z). (33)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 341
V.P. Kotlyarov and E.A. Moskovchenko
P r o o f. The only thing we need is to prove Eq. (32). The matrix M(t, x, z)
defines the solution Φ(t, x, z) of the AKNS Eqs. (8) and (9) by the formula
Φ(t, x, z) = M(t, x, z)e−iθ(t,x,z)σ3 .
Formulas (32) follow from (8) and (RRH3). Indeed, substituting the last formula
into equation (8), we can find that
Mt + iz[σ3,M ] + HM = 0. (34)
Using (RRH3), we put
M(t, x, z) = I +
m(t, x)
z
+ o(z−1),
where
m(t, x) = lim
z→∞ z(M(t, x, z)− I).
This asymptotics and Eq. (34) give
H(t, x) = −i[σ3,m(t, x)], (35)
and hence
E(t, x) = −4im12 = − lim
z→∞ 4izM12(t, x, z).
Further, since M(t, x, z) = m0(t, x) + O(z−1), then the x-equation for M(t, x, z),
Mx − i
4z
Mσ3 = iz[σ3,M ] + HM +
iF
4z
M,
gives F (t, x) = −m0(t, x)σ3m
−1
0 (t, x).
Thus the problem to the Maxwell–Bloch equations, from subsection 2.1 on the
whole t-line and the mixed problem from subsection 2.2 in the quarter xt-plane,
is completely linearizable.
4. More General Matrix Riemann–Hilbert Problems
Now we prove that any Riemann–Hilbert problem like RRH1–RRH3 gene-
rates a solution to the Maxwell–Bloch equations. From here and below we will
consider a more general construction. Let the oriented contour Σ contain a real
line R, sufficiently large circle Γ and some finite arcs γj ∪ γj (j = 1, 2, . . . , p)
which are symmetric with respect to the real line. Thus,
Σ = R ∪ Γ ∪
p⋃
j=1
γj ∪ γj .
342 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations
This type of contours takes place when we deal with periodic initial data or/and
periodic boundary conditions. The contour Σ has the following orientation: the
real line R is oriented from the left to the right, the circle Γ is oriented clockwise,
the arcs γj ∪ γj are oriented up-down. Then the regular matrix RH problem can
be formulated as follows.
Find the 2× 2 matrix M(t, x, z) such that
• M(t, x, z) is analytic in z ∈ C \ Σ and bounded up to the contour Σ; R1
• M−(t, x, z) = M+(t, x, z)J(t, x, z), z ∈ Σ, R2
• M(t, x, z) = I + O(z−1), |z| → ∞. R3
The contour Σ and the jump matrix J(x, t, z) satisfy the Schwartz reflection
principle:
• the contour Σ is symmetric with respect to the real axis R,
• J−1(x, t, z) = J†(x, t, z∗) for z ∈ Σ and Im z 6= 0,
where † and ∗ are Hermitian and complex conjugations, respectively.
• the jump matrix J(x, t, λ) for λ ∈ R has a positive definite real part.
Theorem 4.1. Let the jump matrix J(t, x, z) satisfy the Schwartz reflection
principle and I − J(t, x, .) ∈ L2(Σ) ∩ L∞(Σ). Then for any fixed t, x ∈ R, the
regular RH problem R1, R2, R3 has a unique solution M(t, x, z).
P r o o f. Existence. Let x and t be fixed. We are to find the solution
M(t, x, z) of the RH problem in the form
M(t, x, z) = I +
1
2πi
∫
Σ
P (t, x, s)[I − J(t, x, s)]
s− z
ds, z /∈ Σ. (36)
The Cauchy integral (36) provides all properties of the RH problem (cf.[22]) if
and only if the matrix Q(t, x, λ) := P (t, x, λ) − I satisfies the singular integral
equation
Q(t, x, z)−K[Q](t, x, z) = R(t, x, z), z ∈ Σ. (37)
The singular integral operator K and the right-hand side R(t, x, z) are as follows:
K[Q](t, x, z) :=
1
2πi
∫
Σ
Q(t, x, s)[I − J(t, x, s)]
s− z+
ds,
R(t, x, z) :=
1
2πi
∫
Σ
I − J(t, x, s)
s− z+
ds.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 343
V.P. Kotlyarov and E.A. Moskovchenko
We consider this integral equation in the space L2(Σ) of the 2×2 matrix complex
valued functions Q(z) := Q(t, x, z), z ∈ Σ. The norm of Q ∈ L2(Σ) is given by
||Q||L2(Σ) =
∫
Σ
tr(Q†(z)Q(z))|dz|
1/2
=
2∑
j,l=1
∫
Σ
|Qjl(z))|2|dz|
1/2
.
The operator K is defined by the jump matrix J(t, x, z) and the generalized
function
1
s− z+
= lim
z′→z,z′∈side+
1
s− z′
.
Furthermore, since the jump matrix J(t, x, λ) has a positive definite real part
when λ ∈ R, then Theorem 9.3 from [23] (p. 984) guarantees the L2 invertibility
of the operator Id − K (Id is the identical operator). The function R(t, x, z)
belongs to L2(Σ) because I − J(t, x, z) ∈ L2Σ) when z ∈ Σ, and the Cauchy
operator
C+[f ](z) :=
1
2πi
∫
Σ
f(s)
s− z+
ds =
f(z)
2
+ p.v.
1
2πi
∫
Σ
f(s)
s− z
ds
is bounded in the space L2(Σ) [24]. Therefore, the singular integral Eq. (37) has
a unique solution Q(t, x, z) ∈ L2(Σ) for any fixed x, t ∈ R, and formula (36) gives
the solution of the above RH problem.
Uniqueness. The proof is as follows. Since detJ(t, x, z) ≡ 1, one can find that
detM(t, x, z) ≡ 1 by repeating step by step the proof of Theorem 7.18 from [22]
(p. 194–198). Hence the matrix M−1(t, x, z) exists and it is analytic in z ∈ C\Σ.
Let us now suppose that there is another matrix M̃(t, x, z) which solves the given
Riemann–Hilbert problem. Thus,
M̃−(t, x, z)M−1
− (t, x, z) = M̃+(t, x, z)J(t, x, z)J−1(t, x, z)M−1
+ (t, x, z)
= M̃+(t, x, z)M−1
+ (t, x, z),
and we can find that the matrix M̃(t, x, z)M−1(t, x, z) is analytic in z ∈ C
and it tends to the identity matrix as z → ∞. By Liovilles’s theorem,
M̃(t, x, z)M−1(t, x, z) ≡ I and therefore M̃(t, x, z) ≡ M(t, x, z), i.e., the matrix
M(t, x, z) is unique.
Theorem 4.2. Let M(t, x, z) be the solution of the RH problem (R1)–(R3)
given by Theorem 4.1 with a matrix J(t, x, z) such that
J(t, x, z) = e(−izt+iη(z)x)σ3J0(z)e(izt−iη(z)x)σ3 , η(z) = z − 1
4z
,
344 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations
where J0(z) is independent of t and x. If additionally M(t, x, z) is smooth in t
and x, then the matrix Φ(t, x, z) := M(t, x, z)e(−izt+iη(z)x)σ3 satisfies the AKNS
system of Eqs. (8) and (9) with the functions E(t, x), N (t, x), ρ(t, x) given by
(32) and (33). Moreover, they are smooth and satisfy the MB Eqs. (1)–(3).
P r o o f. The matrix Φ(t, x, z) := M(t, x, z)e(−izt+iη(z)x)σ3 is analytic in
z ∈ C \ Σ and has the jump across Σ:
Φ−(t, x, z) = Φ+(t, x, z)J0(z),
where J0(λ) is independent of t and x. This relation implies:
dΦ−(t, x, z)
dt
Φ−1
− (t, x, z) =
dΦ+(t, x, z)
dt
Φ−1
+ (t, x, z),
dΦ−(t, x, z)
dx
Φ−1
− (t, x, z) =
dΦ+(t, x, z)
dx
Φ−1
+ (t, x, z)
for z ∈ Σ. The last relations mean that the matrix logarithmic derivatives
Φt(t, x, z)Φ−1(t, x, z) and Φx(t, x, z)Φ−1(t, x, z) are analytic in z ∈ C\{0} except
the end points and the points of self intersection of the contour Σ. The matrix
M(t, x, z) and its derivative Mt(t, x, z) (in t) are analytic in z ∈ C\Σ. Moreover,
the Cauchy integral (36) gives the following asymptotic formulas:
M(t, x, z) = I +
m±(t, x)
z
+ O(z−2), z →∞, z ∈ C±.
Hence
Φt(t, x, z)Φ−1(t, x, z) = −izσ3 + i[σ3,m+(t, x)] + O(z−1)
= −izσ3 +i[σ3, m−(t, x)]+O(z−1), z →∞,
where [A,B] := AB −BA and
m−(t, x) = m+(t, x) = m(t, x) =
i
2π
∫
Σ
P (t, x, z)[I − J(t, x, z)]dz.
Since M(t, x, z) is bounded up to the boundary, then z = 0, and the end points
and the points of self-intersection of the contour Σ are removable singularities
for Φt(t, x, z)Φ−1(t, x, z). Therefore, by Liouville’s theorem, this derivative is a
polynomial
U(z) := Φt(t, x, z)Φ−1(t, x, z) = −izσ3 −H(t, x),
where H(t, x) := −i[σ3,m(t, x)] =
(
0 q(t, x)
p(t, x) 0
)
. Using the Schwartz symme-
try properties of the jump matrix J(t, x, z), we can show that U(z) = σ2U
∗(z∗)σ2,
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 345
V.P. Kotlyarov and E.A. Moskovchenko
where σ2 =
(
0 −i
i 0
)
. These reductions imply H(t, x) = −H†(x, t), i.e., q(t, x) =
−p∗(t, x), and we put q(t, x) := E(t, x)/2. Thus Φ(t, x, z) satisfies Eq. (8), and
a scalar function E(t, x) is defined by (32). The function E(t, x) is smooth in t
and x because the matrix M(x, t, z), and hence m(t, x) are smooth in t and x by
supposition. In the same way as before, we can find that Φx(x, t, λ)Φ−1(x, t, λ)
is a rational matrix function
V (z) := Φx(x, t, λ)Φ−1(x, t, λ) = izσ3 + H(t, x) +
iF̂ (t, x)
4z
because the following asymptotics are true:
Φx(t, x, z)Φ−1(t, x, z) = izσ3 + H(t, x) + O(z−1), z →∞,
and
Φx(t, x, z)Φ−1(t, x, z) = − iF̂ (t, x)
4z
+ F0(t, x) + O(z), z → 0,
where F̂ (t, x) = −M(t, x, 0)σ3M
−1(t, x, 0), and F0(t, x) is some matrix. More-
over, the previous relations give: F0(t, x) ≡ H(t, x). Thus the matrix Φ(x, t, z)
satisfies two differential equations:
Φt = U(z)Φ, U(z) = −izσ3 −H(t, x) (38)
Φx = V (z)Φ, V (z) = izσ3 + H(t, x) +
iF̂ (t, x)
4z
. (39)
Their compatibility (Φxt(x, t, λ) = Φtx(x, t, λ)) gives the identity in z,
Ux(z)− Vt(z) + [U(z), V (z)] = 0, [U, V ] = UV − V U,
i.e.,
Ht(t, x) + Hx(t, x) + [izσ3 + H(t, x), izσ3 + H(t, x) +
iF̂ (t, x)
4z
] = 0.
This identity is equivalent to the system of matrix equations:
Ht(t, x) + Hx(t, x) =
1
4
[σ3, F̂ (t, x)] (40)
F̂t(t, x) =[F̂ (t, x),H(t, x)]. (41)
Using the Schwartz symmetry properties of the jump matrix J(t, x, z), we find
that F̂ (t, x) is a Hermitian matrix, and we put
F̂ (t, x) =
(N (t, x) ρ(t, x)
ρ∗(t, x) −N (t, x)
)
.
346 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations
Matrix Eqs. (40) and (41) are equivalent to the scalar Eqs. (6). Thus we have
proved that the matrix Φ(t, x, z) satisfies equations (38) and (39) that coincide
with AKNS system (8) and (9), and the scalar functions E(t, x), N (t, x), ρ(t, x)
are smooth and satisfy the MB equations (6) due to the compatibility of equations
(38) and (39).
As a corollary of the previous theorems, we obtain (due to formulas (32),
(36)) an integral representation for the electric field envelope
E(t, x) =
2
π
∫
Σ
([I + Q(t, x, z)][J(t, x, z)− I])12 dz
through the solution Q(t, x, z) of the singular integral equation (37) which is
equivalent to the regular RH problem. The entries N (t, x) and ρ(t, x) of the
density matrix of a quantum two-level atom subsystem are defined by
(N (t, x) ρ(t, x)
ρ∗(t, x) −N (t, x)
)
= −M(t, x, 0)σ3M
−1(t, x, 0)
or by using linear differential equations (2) and (3) by already known E(t, x).
5. Conclusions
Thus the Riemann–Hilbert problem R1–R3 with the given contour conjuga-
tion and the jump matrix given by Theorem 4.2, which satisfy the Schwartz reflec-
tion principle, generates the solutions to the Maxwell–Bloch equations. Among
them there are the solutions defined for t ∈ R, x ∈ R+ and studied in [5, 6],
the step-like solutions with a different background shape as t → ±∞ that (by
the best of our knowledge) are not considered in the literature, the solutions to
the mixed problem (6), (7) (t, x ∈ R+) with decreasing or periodic input pulse
E(t, 0) and different initial functions E(0, x), N (0, x), ρ(0, x), etc. The type of
solutions is defined by the specialization of the conjugation contour and the jump
matrix on this contour. The specialization which cover the periodicity case and
description of the corresponding solutions will be done further.
Acknowledgments. This research was supported by the grant Network of
Mathematical Research 2013–2015
References
[1] G.L. Lamb Jr., Propagation of Ultrashort Optical Pulses. — Phys. Lett. A 25A
(1967), 181–182.
[2] G.L. Lamb Jr., Analytical Ddescriptions to Ultrashort Optical Pulse Propagation
in Resonant Media. — Rev. Mod. Phys. 43 (1971), 99–124.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 347
V.P. Kotlyarov and E.A. Moskovchenko
[3] G.L. Lamb Jr., Phase Variation in Coherent-optical-pulse Propagation. — Phys.
Rev. Lett. 31 (1973), 196–199.
[4] G.L. Lamb Jr., Coherent-Optical-Pulse Propagation as an Inverse Problem. —
Phys. Rev. A 9 (1974), 422–430.
[5] M.J. Ablowits, D. Kaup, and A.C. Newell, Coherent-Pulse Propagation, a Disper-
sive, Irreversible Phenomenon. — J. Math. Phys. 15 (1974), 1852–1858.
[6] I.R. Gabitov, V.E. Zakharov, and A.V. Mikhailov, Maxwell–Bloch Equations and
Inverse Scattering Transform Method. — Teor. Mat. Fiz. 63 (1985), 11–31.
[7] P. Deift and X. Zhou, A Steepest Descent Method for Oscillatory Riemann–Hilbert
Problems. Asymptotics for the MKdV Equation. — Ann. Math. 137/2 (1993), 295–
368.
[8] V.P. Kotlyarov and A.A. Minakov, Step-Initial Function to the MKdV Equation:
Hyperelliptic Long-Time Asymptotics of the Solution. — J. Math. Phys., Anal.,
Geom. 8 (2012), No. 1, 38–62.
[9] M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform. SIAM,
Philadelphia, 1981.
[10] S.V. Manakov, Propagation of Ultrshort Optical Pulse in a Two-Level Laser Am-
plifier. — Zh. Eksp. Teor. Fiz. 83 (1982), 68–83.
[11] S.V. Manakov and V.Yu. Novokshenov, Complete Asymptotic Representation of
Electromagnetic Pulse in a Long Two-Level Amplifier. — Teor. Mat. Fiz. 69 (1986),
40–54.
[12] O.M. Kiselev, Solution of Goursat Problem for Maxwell–Bloch Equations. — Teor.
Mat. Fiz. 98 (1994), 29–37.
[13] A.S. Fokas, A Unified Transform Method for Solving Linear and Certain Nonlinear
PDEs. — Proc. R. Soc. London, Ser. A 453 (1997), 1411–1443.
[14] A.S. Fokas and A.R. Its, The Linearization of the Initial Boundary Value Problem
of the Nonlinear Schrödinger Equation. — SIAM J. Math. Anal. 27 (1996), 738–764.
[15] A.S. Fokas and A.R. Its, An Initial Boundary Value Problem for the Korteweg de
Vries Equation. — Mathematics and Computer in Simulation 37 (1994), 293–321.
[16] A.S. Fokas and A.R. Its, An Initial Boundary Value Problem for the sine-Gordon
Equation in laboratory coordinates. — Teor. Mat. Fiz. 92 (1992), 387–403.
[17] A. Boutet de Monvel and V.P. Kotlyarov, Scattering Problem for the Zakharov–
Shabat Equations on the Semi-Axis. — Inverse Probl. 16 (2000), 1813–1837.
[18] A. Boutet de Monvel and V.P. Kotlyarov, Generation of Asymptotic Solitons of the
Nonlinear Schrödinger Equation by Boundary Data. — J. Math. Phys. 44 (2003),
3185–3215.
[19] A. Boutet de Monvel and V. Kotlyarov, Focusing Nonlinear Schrödinger Equation
on the Quarter Plane with Time-Periodic Boundary Condition: a Riemann–Hilbert
Approach. — Journal of the Institute of Mathematics of Jussieu 6 (2007), 579–611.
348 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations
[20] E.A. Moskovchenko and V.P. Kotlyrov, A New Riemann–Hilbert Problem in a
Model of Stimulated Raman Scattering. — J. Phys. A: Math. Gen. 39 (2006),
14591–14610.
[21] L.D. Fadeev and L.A. Takhtadjan, Hamiltonian Methods in the Theory of Solitons.
Springer, Berlin, 1987.
[22] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Ap-
proach. CIMS NY University, 1999.
[23] X. Zhou, The Riemann–Hilbert problem and inverse scattering. — SIAM J. Math.
Anal. 20 (1989), 966–986.
[24] G. Litvinchuk and I. Spitkovskii, Factorization of Measurable Matrix Functions.
Birkhäuser–Verlag, Basel, 1987.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 349
|
| id | nasplib_isofts_kiev_ua-123456789-106802 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-07T17:30:51Z |
| publishDate | 2014 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Kotlyarov, V.P. Moskovchenko, E.A. 2016-10-05T19:48:37Z 2016-10-05T19:48:37Z 2014 Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening / V.P. Kotlyarov, E.A. Moskovchenko // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 3. — С. 328-349. — Бібліогр.: 24 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106802 The Maxwell-Bloch equations have been intensively studied by many authors. The main results are based on the inverse scattering transform and the Marchenko integral equations. However this method is not acceptable for mixed problems. In the paper, we develop a method allowing to linearize mixed problems. It is based on simultaneous spectral analysis of both Lax equations and the matrix Riemann{Hilbert problems. We consider the case of infinitely narrow spectral line, i.e., without spectrum broadening. The proposed matrix Riemann-Hilbert problem can be used for studying temporal/spatial asymptotics of the solutions of Maxwell-Bloch equations by using a nonlinear method of steepest descent. Уравнения Максвелла-Блоха интенсивно изучаются многими авторами. Основные результаты базируются на методе обратной задачи с использованием интегральных уравнений Марченко. Однако такой метод оказался неприемлемым для смешанных задач. В данной работе мы развиваем метод, позволяющий линеаризовать смешанные задачи. Он основан на одновременном спектральном анализе обоих уравнений Лакса и матричных задачах Римана-Гильберта. Мы рассматриваем случай бесконечно узкой спектральной линии, т.е. без уширения спектра. Предлагаемые матричные задачи Римана-Гильберта будут полезны для изучения временных/пространственных асимптотик решений уравнений Максвелла-Блоха, используя нелинейный метод наискорейшего спуска. This research was supported by the grant Network of Mathematical Research 2013 - 2015 en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening Article published earlier |
| spellingShingle | Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening Kotlyarov, V.P. Moskovchenko, E.A. |
| title | Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening |
| title_full | Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening |
| title_fullStr | Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening |
| title_full_unstemmed | Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening |
| title_short | Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening |
| title_sort | matrix riemann-hilbert problems and maxwell-bloch equations without spectral broadening |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106802 |
| work_keys_str_mv | AT kotlyarovvp matrixriemannhilbertproblemsandmaxwellblochequationswithoutspectralbroadening AT moskovchenkoea matrixriemannhilbertproblemsandmaxwellblochequationswithoutspectralbroadening |